Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover of X there is a sequence of maps (f n: X → X)nεgw such that each f n is -near to the identity map of X and the family f n(X)n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z ∞-set in X.
@article{bwmeta1.element.doi-10_2478_s11533-008-0005-0,
author = {Taras Banakh and Igor Zarichnyy},
title = {Topological groups and convex sets homeomorphic to non-separable Hilbert spaces},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {77-86},
zbl = {1202.57023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0005-0}
}
Taras Banakh; Igor Zarichnyy. Topological groups and convex sets homeomorphic to non-separable Hilbert spaces. Open Mathematics, Tome 6 (2008) pp. 77-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0005-0/
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